Shock and Vibration

Volume 2018 (2018), Article ID 7303294, 13 pages

https://doi.org/10.1155/2018/7303294

## A New Method for Determining Optimal Regularization Parameter in Near-Field Acoustic Holography

Correspondence should be addressed to Yue Xiao

Received 29 September 2017; Revised 17 January 2018; Accepted 24 January 2018; Published 20 February 2018

Academic Editor: Chao Tao

Copyright © 2018 Yue Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Tikhonov regularization method is effective in stabilizing reconstruction process of the near-field acoustic holography (NAH) based on the equivalent source method (ESM), and the selection of the optimal regularization parameter is a key problem that determines the regularization effect. In this work, a new method for determining the optimal regularization parameter is proposed. The transfer matrix relating the source strengths of the equivalent sources to the measured pressures on the hologram surface is augmented by adding a fictitious point source with zero strength. The minimization of the norm of this fictitious point source strength is as the criterion for choosing the optimal regularization parameter since the reconstructed value should tend to zero. The original inverse problem in calculating the source strengths is converted into a univariate optimization problem which is solved by a one-dimensional search technique. Two numerical simulations with a point driven simply supported plate and a pulsating sphere are investigated to validate the performance of the proposed method by comparison with the L-curve method. The results demonstrate that the proposed method can determine the regularization parameter correctly and effectively for the reconstruction in NAH.

#### 1. Introduction

Near-field acoustic holography (NAH) is an effective technique for noise sources identification and acoustic field visualization. By measurements on a hologram surface near the sound source, the acoustic quantities such as pressure, particle velocity, and intensity anywhere including the source surface can be reconstructed, and the three-dimensional acoustic filed can be predicted. In the past years, many alternative methods of transform algorithm for realizing the NAH have been developed, for example, the spatial Fourier transform [1, 2], the boundary element method [3–5], the Helmholtz equation least squares method [6, 7], the statistically optimized method [8, 9], and the equivalent source method (ESM) [10–14]. Among the methods, the ESM has attracted more attention with the advantages of the simple principle, the high calculation efficiency, and good adaptability for the shapes of the sound sources.

However, the reconstruction of acoustic quantities of sound sources from near-field measurement data is an inverse problem, which is different from the traditional acoustic radiation calculation. Consequently, the reconstruction stability is a key problem in NAH technology due to the fact that the realization process is ill-posed and the reconstructed results are highly sensitive to signal-to-noise ratio (SNR) [15]. If the conventional method for solving inverse matrix from the source to the hologram surface is used directly, the measurement error on the hologram surface as the input may be amplified sharply, and even the reconstructed results are completely unavailable. At present, the regularization approach has been widely used in the reconstruction process to restrict the influence of measurement errors on the reconstructed results and ensure the validity of the reconstruction performance.

Currently, the direct regularization methods and the iterative regularization methods are mostly used for dealing with ill-posed problem of NAH [16, 17]. The direct regularization methods include truncated singular value decomposition (TSVD) method [18] and Tikhonov regularization method [19–21]. The iterative regularization methods include Landweber iterative regularization method [22] and conjugate gradient method [23]. The essence of the regularization method is to find an optimal approximate solution to the actual solution by a certain control strategy. So far there is no single regularization that is absolutely superior to the rest. For each of different regularization methods, the selection criteria of the optimal regularization parameters are the critical factors that determine the regularization effect.

The methods for selection of optimal regularization parameter can be classified into prior [24] and posterior strategy. Compared with prior strategy, the posterior strategy does not require the prior knowledge of the noise variance in advance. The generalized cross validation (GCV) method [25] and the L-curve method [26, 27] as the posterior strategy are the two most widely used methods for selecting the optimal regularization parameters, both of which have good adaptability in the engineering application. Although the convergence of the GCV criterion has been proven theoretically, the GCV function curve sometimes is too flat, which makes it difficult to determine the minimum value of the GCV function. The L-curve method describes the comparison between the solution norm and residual norm with the regularization parameters in a log-log scale coordinate and adopts the maximum curvature of L-curve as the location of the optimal parameter. The L-curve method is reliable for determination of the optimal regularization parameter in most cases, but it may fail to search the maximum curvature position when there is more than one inflexion on the curve or the curve is not L-shaped approximately [28].

This paper focuses on a new method for determining the optimal parameters of Tikhonov regularization in NAH technique based on the ESM. In the proposed method, the transfer matrix from the source to the hologram surface is updated by augmenting a fictitious point source with zero strength near the locations of the equivalent sources. The minimization of the norm of this fictitious point source strength can be as the criterion for selection of the optimal regularization parameter since the reconstructed value should tend to zero. Thus the original inverse problem is converted into a univariate optimization problem that can be solved by one-dimensional search technique. The numerical simulations are investigated to demonstrate the validity of the proposed method, and the results show that the novel proposed method is able to select the regularization parameter correctly and effectively.

#### 2. NAH Based on ESM

The basic idea of the NAH based on ESM is that the actual radiated sound field can be replaced by superposing the acoustic field produced by a number of equivalent sources distributed inside the source surface. The source strengths of the equivalent sources can be evaluated by matching the acoustic pressures measured on the hologram surface, and the acoustic quantities in the acoustic field can be obtained by the source strengths and the transfer matrices constructed by the equivalent sources.

As shown in Figure 1, the hologram surface is placed near the outside of the source surface , and the equivalent source surface is located on the other side of the source surface . Suppose that there are equivalent sources on the surface , and the acoustic pressure at the field point can be expressed aswhere is the acoustic pressure at the field point , is the density of the medium, is the speed of sound, is the wave number, is the wavelength, is the angular frequency, and are the location vector and the strength of the th equivalent source, respectively, and is the free field Green’s function obtained from the expression